Which curve shape is considered ideal for vertical alignment (sag and summit) in highway design?

Introduction / Context:
Vertical curves connect grades to provide smooth vehicle motion, adequate sight distance, ride comfort, and effective drainage. The curve geometry must yield a constant rate of change of grade to minimize jerk and ensure predictable sight lines.

Given Data / Assumptions:

• We seek the ideal geometric form for vertical curves used on highways.
• Applies to both summit and sag vertical curves.

Concept / Approach:
A parabola y = ax^2 + bx + c provides a constant rate of change of gradient (second derivative is constant). This property ensures comfortable transition of acceleration and predictable headlight/sight envelope behavior, making parabolas standard for vertical alignment design.

Step-by-Step Solution:
Compare curve types for vertical use: parabola offers constant grade change; spirals/lemniscates are mostly for horizontal transitions.Parabolic curve ensures uniform rate of change of slope (dv/dx constant).Therefore, select Parabola as the ideal vertical curve.

Verification / Alternative check:
Design manuals and standard drawings adopt parabolic equations for computing lengths based on sight distance and comfort criteria, confirming industry practice.

Why Other Options Are Wrong:

• Clothoid/Cubic spiral/Lemniscate: used primarily for horizontal transition to vary curvature, not for vertical.
• Circular arc: produces non-uniform rate of change of grade, causing discomfort and non-linear sight envelopes.

Common Pitfalls:

• Assuming the same transition curves used horizontally are also ideal vertically.
• Using circular arcs in vertical alignment can lead to sight distance inconsistencies.

Final Answer:
Parabola